Conservation laws, soliton solutions for modified Camassa–Holm equation and (2 + 1)-dimensional ZK–BBM equation

Elboree, Mohammed K.

doi:10.1007/s11071-017-3640-9

Cited by 12 publications

(8 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Elboree [22] the conservation laws were constructed in addition to the corresponding conserved quantities for the modified Camassa-Holm equation and (2+1) dimensional Zakharov-Kuznetsov -Benjamin-Bona-Mahoney equation, in this work we have made a preliminary test of conservation of our solutions eqs. (14,29,34) such that, for the solution by factorization, eq.…”

Section: Discussionmentioning

confidence: 99%

Untitled

2021

AJMS

View full text Add to dashboard Cite

In this work, we study the Camassa Holm Degasperis Procesi equation through solving it by three different methods which are the factorization technique, the Cole-Hopf method, and the Schwarzian derivatives method, plus studying the stability of the system through the Phase Portrait method. Illustrations of the solution are presented using symbolic software which shows different pattern formations.

show abstract

Section: Discussionmentioning

confidence: 99%

Untitled

2021

AJMS

View full text Add to dashboard Cite

show abstract

“…where k is a constant. As the typical shallow water wave equation, the Camassa-Holm equation has attracted significant attention in the last two decades because of its interesting properties which including the complete integrability, the presence of breaking waves, and algebro-geometric formulations [28,29]. An interesting fact of this nonlinearly dispersive equation is that it supports the peakon solutions.…”

Section: Introductionmentioning

confidence: 99%

Mastering the Cahn–Hilliard equation and Camassa–Holm equation with cell-average-based neural network method

et al. 2022

View full text Add to dashboard Cite

In this paper, we develop cell-average based neural network (CANN) method to approximate solutions of nonlinear Cahn-Hilliard equation and Camassa-Holm equation. The CANN method is motivated by the finite volume scheme and evolved from the integral or weak formulation of partial differential equations. The major idea of cell-average based neural network method is to explore a neural network to approximate the solution average difference or evolution between two neighboring time steps. Unlike traditional numerical methods, CANN method is not limited by the CFL restriction and can adapt large time steps for solution evolution, which is a significant advantage that classical numerical methods do not have. Once well trained, this method can be implemented as an fixed explicit finite volume scheme and applied to certain groups of initial value conditions for Cahn-Hilliard equation and Camassa-Holm equation without retraining the neural network. Furthermore, the CANN method also performs very well in handling data with corruption or low quality data generated by Gaussian white noise. Numerical examples are presented to demonstrate effectiveness, accuracy and capability of the proposed method.

show abstract

“…In Section 2, the semi-inverse method and the variational approach are used to derive the (4+1)-dimensional time fractional Fokas equation. In Section 3, we make use of Lie symmetry analysis to study the symmetry of the time-fractional equation, and we construct the conservation laws of the equation by the new conservation theorem [38][39][40][41][42][43][44]. In Section 4, the exact solutions of the time fractional equation are given by using the G G -expansion method.…”

Section: Introductionmentioning

confidence: 99%

Analytical and Numerical Solutions for a Kind of High-Dimensional Fractional Order Equation

Hou

Zhang

2022

Fractal Fract

View full text Add to dashboard Cite

In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. It is rarely studied because the order of the highest-order derivative term of this equation is higher than the common generalized (4+1)-dimensional Fokas equation. Firstly, the (4+1)-dimensional time-fractional Fokas equation with the Riemann–Liouville fractional derivative is derived by the semi-inverse method and variational method. Further, the symmetry of the time-fractional equation is obtained by the fractional Lie symmetry analysis method. Based on the symmetry, the conservation laws of the time fractional equation are constructed by the new conservation theorem. Then, the G′G-expansion method is used here to solve the equation and obtain the exact traveling wave solutions. Finally, the spectral method in the spatial direction and the Gru¨nwald–Letnikov method in the time direction are considered to obtain the numerical solutions of the time-fractional equation. The numerical solutions are compared with the exact solutions, and the error results confirm the effectiveness of the proposed numerical method.

show abstract

Conservation laws, soliton solutions for modified Camassa–Holm equation and (2 + 1)-dimensional ZK–BBM equation

Cited by 12 publications

References 41 publications

Untitled

Untitled

Mastering the Cahn–Hilliard equation and Camassa–Holm equation with cell-average-based neural network method

Analytical and Numerical Solutions for a Kind of High-Dimensional Fractional Order Equation

Contact Info

Product

Resources

About